| DataSet | Proximity Criterion | Deer | Observations |
|---|---|---|---|
| 1 | closest in time | 35 | 149 |
| 2 | nearest | 35 | 147 |
| 3 | score | 36 | 223 |
P15.2 Fortgeschrittenes Praxisprojekt
Dr. Nicolas Ferry - Bavarian National Forest Park / Daniel Schlichting - StaBLab
31 Jan 2025
Model FCM levels on spatial and temporal distance to hunting activities
Expectations:
Contains information of 809 faecal samples, including:
Samples where taken at irregular time intervals from 2020 to 2022.
Other sources of uncertainty include:
lack of information about hunting events (single time points as start, end, middle?)
unknown characteristics of the deer (e.g., age, health, etc.),
other unknown stressors (e.g., predators, human activities, weather, etc.),
unknown geographical features (e.g., terrain could affect the propagation of sound).
Deer location at the time of hunting event is approximated by linear interpolation:
A hunting event is considered relevant to a faecal sample, if
In this presentation:
Among the relevant hunting events, the most relevant one is defined by one the three proximity criteria:
we define the Scoring function as following:
\[ S(d, t) \propto \begin{cases} \frac{1}{d^2} \cdot f_\textbf{t}(t), t \sim \mathcal{N}(\mu, \sigma^2) &|t \leq \mu \\ \frac{1}{d^2} \cdot f_\textbf{t}(t), t \sim \mathcal{Laplace}(\mu, b) &|t > \mu \end{cases} \] where:
\[ \begin{align*} d & \text{: Distance } \\ t & \text{: Time Difference } \\ \mu & \text{: GRT target = 19 hours } \end{align*} \]
The marginal effects of distance and elapsed time since challenge on the score:
We report models fitted on the following datasets:
| DataSet | Proximity Criterion | Deer | Observations |
|---|---|---|---|
| 1 | closest in time | 35 | 149 |
| 2 | nearest | 35 | 147 |
| 3 | score | 36 | 223 |
We chose two different approaches to Modelling:
For Modelling, we consider the following covariates, defined for each pair of FCM sample and most relevant hunting event:
Family: Gamma
Let \(i = 1,\dots,N\) be the indices of deer and \(j = 1,\dots,n_i\) be the indices of faecal samples for each deer
\[ \begin{eqnarray} \textup{FCM}_{ij} &\overset{\mathrm{iid}}{\sim}& \mathcal{Ga}\left( \nu, \frac{\nu}{\mu_{ij}} \right) \quad\text{for}\; j = 1,\dots,n_i, \\ \mu_{ij} &=& \mathbb{E}(\textup{FCM}_{ij}) = \exp(\eta_{ij}), \\ \eta_{ij} &=& \beta_0 + \beta_1 \cdot \textup{number of other relevant hunting events}_{ij} + \\ && f_1(\textup{time difference}_{ij}) + f_2(\textup{distance}_{ij}) + \\ && f_3(\textup{sample delay}_{ij}) + f_4(\textup{defecation day}_{ij}) + \\ && \gamma_{i}, \\ \gamma_i &\overset{\mathrm{iid}}{\sim}& \mathcal{N}(0, \sigma_\gamma^2) \end{eqnarray} \]
\(f_1, f_2, f_3, f_4\) are penalized cubic regression splines.
High uncertainty (large standard error) about estimated effects, in particular of time difference and distance, across all datasets.
Consistent pattern of sample delay effect when using REML method: larger sample delay \(\Rightarrow\) lower FCM level, as expected.
Instability with respect to estimation methods. GCV tends to yield more wiggly smooth effects than REML.
Estimation of random intercepts is sensitive to choice of dataset.
across all three Models:
| Model | Mean RMSE | SD RMSE | Number of Observations |
|---|---|---|---|
| last | 168.63 | 24.41 | 149 |
| nearest | 151.32 | 17.92 | 147 |
| score | 147.98 | 16.50 | 223 |
| Method | Dataset | Estimate | exp(Estimate) | Standard error |
|---|---|---|---|---|
| REML | Closest in Time | -0.09 | 0.91 | 0.06 |
| REML | Nearest | -0.07 | 0.93 | 0.06 |
| REML | Highest Score | -0.01 | 0.99 | 0.01 |
| GCV | Closest in Time | -0.14 | 0.87 | 0.06 |
| GCV | Nearest | -0.10 | 0.90 | 0.06 |
| GCV | Highest Score | -0.02 | 0.98 | 0.01 |
We do this seperately for all 3 datasets (nearest, closest and score).
Effect of Hunting on Red Deer